Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 12 (2016), 026, 19 pages
Basic Forms and Orbit Spaces:
a Dif feological Approach
Yael KARSHON
†
and Jordan WATTS
‡
†
Department of Mathematics, University of Toronto,
40 St. George Street, Toronto Ontario M5S 2E4, Canada
E-mail: karshon@math.toronto.edu
URL: http://www.math.toronto.edu/karshon/
‡
Department of Mathematics, University of Colorado Boulder,
Campus Box 395, Boulder, CO, 80309, USA
E-mail: jordan.watts@colorado.edu
URL: http://euclid.colorado.edu/~jowa8403
Received October 06, 2015, in final form February 16, 2016; Published online March 08, 2016
http://dx.doi.org/10.3842/SIGMA.2016.026
Abstract. If a Lie group acts on a manifold freely and properly, pulling back by the quotient
map gives an isomorphism between the differential forms on the quotient manifold and the
basic differential forms upstairs. We show that this result remains true for actions that are
not necessarily free nor proper, as long as the identity component acts properly, where on
the quotient space we take differential forms in the diffeological sense.
Key words: diffeology; Lie group actions; orbit space; basic differential forms
2010 Mathematics Subject Classification: 58D19; 57R99
1
Introduction
Let M be a smooth manifold and G a Lie group acting on M . A basic differential form on M
is a differential form that is G-invariant and horizontal; the latter means that evaluating the
form on any vector that is tangent to a G-orbit yields 0. Basic differential forms constitute
a subcomplex of the de Rham complex. If G acts properly and with a constant orbit-type, then
the quotient M/G is a manifold, and, denoting the quotient map by π : M → M/G, the pullback
by this map gives an isomorphism of the de Rham complex on M/G with the complex of basic
forms on M . Even if M/G is not a manifold, if G acts properly, then the cohomology of the
complex of basic forms is isomorphic to the singular cohomology of M/G with real coefficients;
this was shown by Koszul in 1953 [17] for compact group actions and by Palais in 1961 [19] for
proper group actions. In light of these facts, some authors define the de Rham complex on M/G
to be the complex of basic forms on M .
There is another, intrinsic, definition of a differential form on M/G, which comes from
viewing M/G as a diffeological space (see Section 2). This definition agrees with the usual one
when M/G is a manifold. Differential forms on diffeological spaces admit exterior derivatives,
wedge products, and pullbacks under smooth maps. Spaces of differential forms are themselves
diffeological spaces too. With this notion, here is our main result:
Theorem 1.1. Let G be a Lie group acting on a manifold M . Let π : M → M/G be the quotient
map.
(i) The pullback map π ∗ : Ω∗ (M/G) → Ω∗ (M ) is one-to-one. Its image is contained in the
space Ω∗basic (M ) of basic forms. As a map to its image, the map π ∗ is an isomorphism of
differential graded algebras and a diffeological diffeomorphism.
2
Y. Karshon and J. Watts
(ii) If the restriction of the action to the identity component of G is proper, then the image of
the pullback map is equal to the space of basic forms:
π ∗ : Ω∗ (M/G)
∼
=
/ Ω∗ (M ) .
basic
We prove Theorem 1.1 in Section 5, after the proof of Proposition 5.10.
Remark 1.2.
1. Part (i) of Theorem 1.1, which follows from the results of Sections 2 and 3, is not difficult.
The technical heart of Theorem 1.1 is Part (ii), which is proved in Proposition 5.10: if the
identity component of the group acts properly, then every basic form on M descends to
a diffeological form on M/G. This fact is non-trivial even when the group G is finite.
2. The quotient M/G can be non-Hausdorff. Nevertheless, even if its topology is trivial, M/G
may have non-trivial differential forms, and its de Rham cohomology may be non-trivial.
See, for example, the irrational torus in Remark 5.12.
3. We expect the conclusion of Part (ii) of Theorem 1.1 to hold under more general hypotheses. In particular, our assumption that the identity component of G act properly on M is
sufficient but not necessary; see, for instance, Example 5.13.
Diffeology was developed by Jean-Marie Souriau (see [26]) around 1980, following earlier work
of Kuo-Tsai Chen (see, e.g., [4, 5]). Our primary reference for this theory is the book [14] by
Iglesias-Zemmour. In many applications, diffeology can serve as a replacement for the manifold
structures (modelled on locally convex topological vector spaces) on spaces of smooth paths,
functions, or differential forms. See, for example, [2].
The category of diffeological spaces is complete and co-complete (see [1]); in particular,
subsets and quotients naturally inherit diffeological structures. It is also Cartesian closed, where
we equip spaces of smooth maps with their natural functional diffeology.
It is also common to consider M/G as a (Sikorski) differential space, by equipping it with
the set of those real valued functions whose pullback to M is smooth. See Hochschild [12],
Bredon [3], G. Schwarz [22], and Cushman and Śniatycki [7]. This structure is determined by
the diffeology on M/G but is weaker. For example, the quotients Rn /SO(n) for different positive
integers n are isomorphic as differential spaces but not as diffeological spaces. (See Exercise 50
of Iglesias [14] with solution at the end of the book.) There are several inequivalent notions of
“differential form” on differential spaces (see Śniatycki [25] and Watts [27]); we do not know of
an analogue of Theorem 1.1 for any of these notions.
Turning to higher category theory, we can also consider the stack quotient [M/G]. It is
a differentiable stack over the site of manifolds, and it is represented by the action groupoid
G × M ⇒M . One can define a differential k-form on the stack [M/G] as a map of stacks from
[M/G] to the stack of differential forms Ωk . In [30], Watts and Wolbert define a functor Coarse
from stacks to diffeological spaces for which Coarse([M/G]) is equal to M/G equipped with
the quotient diffeology. In this language, Theorem 1.1 gives an isomorphism
Ωk ([M/G]) ∼
= Ωk (Coarse([M/G])),
when the identity component of G acts properly. We note that a diffeological space can also be
viewed as a stack, and applying Coarse recovers the original diffeological space. We also note
that the quotient stack [M/G] often contains more information than the quotient diffeology;
for example, if M is a point, the quotient diffeological space is a point, but the stack [M/G]
determines the group G. Finally, we note that Karshon and Zoghi [16] give sufficient conditions
Basic Forms and Orbit Spaces: a Diffeological Approach
3
for a Lie groupoid to be determined up to Morita equivalence by its underlying diffeological
space.
In the special case that the Lie group G is compact, the main results of this paper appeared
in the Ph.D. Thesis of the second author [28], supervised by the first author. A generalisation
to proper Lie groupoids, which relies on these results and on a deep theorem of Crainic and
Struchiner [6] showing that all proper Lie groupoids are linearisable, was worked out by Watts
in [29].
The paper is structured as follows. For the convenience of the reader, Section 2 contains
background on diffeology, differential forms on diffeological spaces, and Lie group actions in
connection to diffeology. Section 3 contains a proof that the pullback map from the space of
differential forms on the orbit space M/G to the space of differential forms on the manifold M
is an injection into the space of basic forms, and is a diffeomorphism onto its image. Section 4
contains a technical lemma: “quotient in stages”. Section 5 is the technical heart of the paper;
it contains a proof that, if the identity component of the group acts properly, then the pullback
map surjects onto the space of basic forms.
Our appendices contain two applications of the special case of Theorem 1.1 when the group G
is finite. In Appendix A we show that, on an orbifold, the notion of a diffeological differential
form agrees with the usual notion of a differential form on the orbifold. In Appendix B we show
that, on a regular symplectic quotient (which is also an orbifold), the notion of a diffeological
differential form also agrees with Sjamaar’s notion of a differential form on the symplectic
quotient. The case of non-regular symplectic quotients is open.
2
Background on dif feological spaces
In this section we review the basics of diffeology, diffeological differential forms, and Lie group
actions in the context of diffeology. For more details (e.g., the quotient diffeology is in fact
a diffeology), see Iglesias-Zemmour [14].
The basics of dif feology
This subsection contains a review of the basics of diffeology; in particular, the definition of
a diffeology and diffeologically smooth maps, as well as various constructions in the diffeological
category.
Definition 2.1 (diffeology). Let X be a set. A parametrisation on X is a function p : U → X
where U is an open subset of Rn for some n. A diffeology D on X is a set of parametrisations
that satisfies the following three conditions.
1. (Covering) For every point x ∈ X and every non-negative integer n ∈ N, the constant
function p : Rn → {x} ⊆ X is in D.
2. (Locality) Let p : U → X be a parametrisation such that for every point in U there exists
an open neighbourhood V in U such that p|V ∈ D. Then p ∈ D.
3. (Smooth compatibility) Let (p : U → X) ∈ D. Then for every n ∈ N, every open subset
V ⊆ Rn , and every smooth map F : V → U , we have p ◦ F ∈ D.
A set X equipped with a diffeology D is called a diffeological space and is denoted by (X, D).
When the diffeology is understood, we drop the symbol D. The elements of D are called plots.
Example 2.2 (standard diffeology on a manifold). Let M be a manifold. The standard diffeology
on M is the set of all smooth maps to M from open subsets of Rn for all n ∈ N.
4
Y. Karshon and J. Watts
Definition 2.3 (diffeologically smooth maps). Let X and Y be two diffeological spaces, and
let F : X → Y be a map. We say that F is (diffeologically) smooth if for any plot p : U → X
of X the composition F ◦ p : U → Y is a plot of Y . Denote by C ∞ (X, Y ) the set of all smooth
maps from X to Y . Denote by C ∞ (X) the set of all smooth maps from X to R, where R is
equipped with its standard diffeology.
Remark 2.4 (plots). A parametrisation is diffeologically smooth if and only if it is a plot.
Remark 2.5 (smooth maps between manifolds). A map between two manifolds is diffeologically
smooth if and only if it is smooth in the usual sense. In particular, if M is a manifold then
C ∞ (M ) is the usual set of smooth real valued functions.
Remark 2.6. Diffeological spaces, along with diffeologically smooth maps, form a category. It
is shown in [1, Theorem 3.2] that this category is a complete and cocomplete quasi-topos. In
particular, it is closed under passing to arbitrary quotients, subsets, function spaces, products,
and coproducts.
Definition 2.7 (quotient diffeology). Let X be a diffeological space, and let ∼ be an equivalence
relation on X. Let Y = X/∼ be the quotient set, and let π : X → Y be the quotient map. We
define the quotient diffeology on Y to be the diffeology for which the plots are those maps
p : U → Y such that for every point in U there exist an open neighbourhood V ⊆ U and a plot
q : V → X such that p|V = π ◦ q.
Remark 2.8 (quotient map). Let X be a diffeological space and ∼ an equivalence relation
on X. Then the quotient map π : X → X/∼ is smooth.
A special case that is important to us is the quotient of a manifold by the action of a Lie
group.
Definition 2.9 (subset diffeology). Let X be a diffeological space, and let Y be a subset of X.
The subset diffeology on Y consists of those maps to Y whose composition with the inclusion
map Y → X are plots of X.
Definition 2.10 (product diffeology). Let X and Y be two diffeological spaces. The product
diffeology on the set X × Y is defined as follows. Let prX : X × Y → X and prY : X × Y → Y
be the natural projections. A parametrisation p : U → X × Y is a plot if prX ◦p and prY ◦p are
plots of X and Y , respectively.
Definition 2.11 (standard functional diffeology on maps). Let Y and Z be diffeological spaces.
The standard functional diffeology on C ∞ (Y, Z) is defined as follows. A parametrisation p : U →
C ∞ (Y, Z) is a plot if the map
U ×Y →Z
given by
(u, y) 7→ p(u)(y)
is smooth.
Dif feological dif ferential forms
In this subsection we review differential forms on diffeological spaces, as well as introduce
Proposition 2.20 which is a simple criterion crucial to the proof of Theorem 1.1.
Definition 2.12 (differential forms). Let (X, D) be a diffeological space. A (diffeological)
differential k-form α on X is an assignment to each plot (p : U → X) ∈ D a differential k-form
α(p) ∈ Ωk (U ) satisfying the following smooth compatibility condition: for every open subset V
of a Euclidean space and every smooth map F : V → U ,
α(p ◦ F ) = F ∗ (α(p)).
Denote the set of differential k-forms on X by Ωk (X).
Basic Forms and Orbit Spaces: a Diffeological Approach
5
Definition 2.13 (pullback map). Let X and Y be diffeological spaces, and let F : X → Y be
a diffeologically smooth map. Let α be a differential k-form on Y . Define the pullback F ∗ α to
be the k-form on X that satisfies the following condition: for every plot p : U → X,
F ∗ α(p) = α(F ◦ p).
Example 2.14. Let α be a differential form on a manifold M . Then (p : U → M ) 7→ p∗ α
defines a diffeological differential form on M . In this way, we get an identification of the
ordinary differential forms on M with the diffeological differential forms on M .
Let X be a diffeological space, α a differential form on X, and p : U → X a plot. The above
identification of ordinary differential forms on U with diffeological differential forms on U gives
α(p) = p∗ α. Henceforth, we may write p∗ α instead of α(p).
Example 2.15. The space Ω0 (X) of diffeological 0-forms is identified with the space C ∞ (X) of
smooth real valued functions, by identifying the function f with the 0-form (p : U → X) 7→ f ◦ p.
See [14, Section 6.31]. With this identification, the pullback of 0-forms by a smooth map F
becomes the precomposition of smooth real-valued functions by F .
Remark 2.16 (pullback is linear). The space of differential forms on a diffeological space is
naturally a linear vector space: for α, β ∈ Ωk (X) and a, b ∈ R, we define aα + bβ : (p : U →
X) 7→ aα(p) + bβ(p). If F : X → Y is a smooth map of diffeological spaces, then the pullback
map F ∗ : Ωk (Y ) → Ωk (X) is linear.
Remark 2.17 (wedge product and exterior derivative). Let X be a diffeological space. Define
the wedge product of α ∈ Ωk (X) and β ∈ Ωl (X) to be the (k + l)-form α ∧ β : (p : U → X) 7→
p∗ α ∧ p∗ β. Define
derivative of α to be the k + 1 form dα : (p : U → X) 7→ d(p∗ α).
L∞the exterior
k
∗
Then Ω (X) = k=0 Ω (X) is a differential graded algebra. In particular, Ω∗ (X) is an exterior
algebra, and (Ω∗ (X), d) is a complex. If F : X → Y is a smooth map of diffeological spaces,
then the pullback map F ∗ : Ω∗ (Y ) → Ω∗ (X) is a morphism of differential graded algebras; in
particular, it intertwines the wedge products and the exterior derivatives.
Definition 2.18 (standard functional diffeology on forms). Let (X, D) be a diffeological space.
The standard functional diffeology on Ωk (X) is defined as follows. A parametrisation p :VU →
Ωk (X) is a plot if for every plot (q : V → X) ∈ D, where V is open in Rn , the map U ×V → k Rn
sending (u, v) to q ∗ (p(u))|v is smooth. See [14, Section 6.29] for a proof that this is indeed
a diffeology.
Remark 2.19. Let X be a diffeological space. We have the following facts.
1. Under the identification of Example 2.15 of the space of diffeological 0-forms with the
space of smooth real valued functions, Definitions 2.11 and 2.18 of the standard functional
diffeology on these spaces agree.
2. If F : X → Y is a smooth map to another diffeological space, then the pullback map
F ∗ : Ωk (Y ) → Ωk (X) is smooth with respect to the standard functional diffeologies on the
sets of differential forms. See [14, Section 6.32].
3. The exterior derivative d : Ωk (X) → Ωk+1 (X) and the wedge product Ωk (X) × Ωl (X) →
Ωk+l (X) are smooth. See Sections 6.34 and 6.35 of [14].
Proposition 2.20 (pullbacks of quotient diffeological forms). Let G be a Lie group, acting on
a manifold M , and let π : M → M/G be the quotient map. Then a differential form α on M
is in the image of π ∗ if and only if, for every two plots p1 : U → M and p2 : U → M such that
π ◦ p1 = π ◦ p2 , we have
p∗1 α = p∗2 α.
Proof . This result is a special case of [14, Section 6.38].
6
Y. Karshon and J. Watts
Group actions
Here we highlight some facts about Lie group actions in connection to diffeology, as well as
review the definition of basic forms. Finally, we introduce another crucial ingredient to the
proof of Theorem 1.1, the slice theorem (Theorem 2.25). Koszul [17] proved the slice theorem
for compact Lie group actions, and Palais [19] proved it for proper Lie group actions. The proof
is also described in Theorem 2.3.3 of [9] and in Appendix B of [10].
Lemma 2.21. Let a Lie group G act on a manifold M . Let x be a point in M and H its
stabiliser. Then
• There exists a unique manifold structure on the quotient G/H such that the quotient map
G → G/H is a submersion. The standard diffeology on this manifold agrees with the
quotient diffeology induced from G.
• There exists a unique manifold structure on the orbit G · x such that the inclusion map
G · x → M is an immersion. The standard diffeology on this manifold agrees with the
subset diffeology induced from M .
• The orbit map a 7→ a · x from G to M descends to a diffeomorphism from G/H to G · x.
• The tangent space Tx (G · x) is the space of vectors ξM |x for ξ ∈ g, where ξM is the vector
field on M that is induced by the Lie algebra element ξ. This space is also the image of
the differential at the identity of the orbit map a 7→ a · x from G to M .
Proof . See [15, Section 2, Paragraph 1].
Definition 2.22 (basic forms). Let G be a Lie group acting on a manifold M . A differential
form α on M is horizontal if for any x ∈ M and v ∈ Tx (G · x) we have
vy α = 0.
(Recall that vy α = α(v, ·, . . . , ·).) A form that is both horizontal and G-invariant is called basic.
When the G-action is understood, we denote the set of basic k-forms on M by Ωkbasic (M ).
Remark 2.23. The space of basic differential forms on a G-manifold M is closed under linear
combinations, wedge products, and exterior derivatives.
Remark 2.24. Given a quotient map π : X → X 0 of diffeological spaces, (more generally,
given a so-called subduction,) Iglesias-Zemmour [14, Section 6.38] defines a “basic form” to be
a differential form on X that satisfies the technical condition that appears in Proposition 2.20
above. Our results show that, for smooth Lie group actions where the identity component acts
properly, Iglesias-Zemmour’s definition agrees with the usual one.
Let G be a Lie group, H a closed subgroup, and V a vector space with a linear H-action.
The equivariant vector bundle
G ×H V
over G/H is obtained as the quotient of G × V by the anti-diagonal H-action h · (g, v) =
(gh−1 , h · v). The G-action on G ×H V is g · [g 0 , v] = [gg 0 , v].
Theorem 2.25 (slice theorem). Let G be a Lie group acting properly on a manifold M . Fix
x ∈ M . Let H be the stabiliser of x, and let V = Tx M/Tx (G · x) be the normal space to the
orbit G · x at x, equipped with the linear H-action that is induced by the linear isotropy action
of H on Tx M . Then there exist a G-invariant open neighbourhood U of x and a G-equivariant
diffeomorphism F : U → G ×H V that takes x to [1, 0].
Basic Forms and Orbit Spaces: a Diffeological Approach
3
7
The pullback injects into basic forms
In this section we prove the easy part of Theorem 1.1: for a Lie group G acting on a manifold M
with quotient map π : M → M/G, the pullback map π ∗ is an injection from the set of diffeological
differential forms on M/G into the set of basic forms on M , and π ∗ is a diffeomorphism onto
its image.
The following lemma is a special case of [14, Section 6.39].
Lemma 3.1. Let a Lie group G act on a manifold M , and let π : M → M/G be the quotient
map. Then the pullback map on forms, π ∗ : Ωk (M/G) → Ωk (M ), is an injection.
Proof . More generally, let X be a diffeological space, ∼ an equivalence relation on X, and
π : X → X/∼ the quotient map. Then the pullback map on forms, π ∗ : Ωk (X/∼) → Ωk (X), is
an injection.
By Remark 2.16 it is enough to show that the kernel of the pullback map
π ∗ : Ωk (X/∼) → Ωk (X)
is trivial. Let α ∈ Ωk (X/ ∼) be such that π ∗ α = 0. Then, for any plot p : U → X, we
have p∗ π ∗ α = 0. By the definition of the quotient diffeology, this implies that for any plot
q : U → X/∼ we have q ∗ α = 0. Hence, α = 0, as required.
Proposition 3.2 (pullbacks from the orbit space are basic). Let a Lie group G act on a manifold M . Let α = π ∗ β for some β ∈ Ωk (M/G). Then α is basic.
Proof . To show that α is G-invariant, note that for every g ∈ G, because π ◦ g = π,
g ∗ α = g ∗ π ∗ β = π ∗ β = α.
If α is a zero-form (that is, a smooth function) then α is automatically horizontal and we are
done. Next, we assume that α is a differential form of positive degree, and we show that α is
horizontal. By Lemma 2.21, if x ∈ M and v ∈ Tx (G · x), then there exists ξ ∈ g such that
d v = exp(tξ) · x.
dt t=0
Let Ax : G → M be the map sending g to g · x. Then, v = (Ax )∗ (ξ).
Thus, to show that α is horizontal, it is enough to show that A∗x α = 0 for all x ∈ M . Indeed,
it follows from the following commutative diagram that A∗x α = A∗x π ∗ β = 0,
G
g7→?
Ax
{?}
/M
?7→[x]
π
/ M/G
Remark 3.3. Proposition 3.2 can also be deduced from the following lemma: a differential
form α on M is basic if and only if its pullbacks under the maps G × M → M given by the
projection (g, x) 7→ x and by the action (g, x) 7→ g · x coincide. For a proof of this lemma see,
for example, [29, Lemma 3.3].
Proposition 3.4 (pullbacks via quotient maps). Let a Lie group G act on a manifold M , and
let π : M → M/G be the quotient map. Then the pullback map
π ∗ : Ωk (M/G) → π ∗ Ωk (M/G)
is a diffeomorphism, where the target space is equipped with the subset diffeology induced
from Ωk (M ).
8
Y. Karshon and J. Watts
Proof . More generally, let X be a diffeological space, let ∼ be an equivalence relation on X,
and let π : X → X/∼ be the quotient map. We will show that the pullback map
π ∗ : Ωk (X/∼) → π ∗ Ωk (X/∼)
is a diffeomorphism, where the target space is equipped with the subset diffeology induced
from Ωk (X).
Clearly, π ∗ is surjective to its image. For the injectivity of π ∗ , see Lemma 3.1. As noted in
Part (2) of Remark 2.19, π ∗ is smooth. We wish to show that the inverse map
(π ∗ )−1 : π ∗ Ωk (X/∼) → Ωk (X/∼)
is also smooth.
Fix a plot p : U → Ωk (X) with image in π ∗ Ωk (X/∼). We would like to show that (π ∗ )−1 ◦
p : U → Ωk (X/ ∼) is a plot of Ωk (X/ ∼). By Definition 2.18 of the diffeology on spaces of
differential forms, we need to show, given any plot r : W → X/∼ with W ⊂ Rn , that
(u, w) 7→ r∗ (π ∗ )−1 ◦ p (u) w
V
is a map to k Rn that is smooth in (u, w) ∈ U × W . Here, (π ∗ )−1 is restricted to the image
of π ∗ , on which it is well defined because π ∗ is injective.
It is enough to show smoothness locally. For any point w ∈ W there exist an open neighbourhood V ⊆ W of w and a plot q : V → X such that r|V = π ◦ q. For all v ∈ V , we
have
r∗ (π ∗ )−1 ◦ p (u) |v = q ∗ (p(u))|v ,
which is smooth in (u, v) ∈ U × V by the definition of the standard functional diffeology
on Ωk (X). And so we are done.
4
Quotient in stages
In this section we give a technical result that we use in the next section. All quotients, subsets,
and products are assumed to be equipped with the quotient, subset, and product diffeologies.
On the image of any Lie group homomorphism H → G there exists a unique manifold
structure such that the inclusion map of the image into G is an immersion; this follows from
the second item of Lemma 2.21. By Lie subgroup of G we refer to such an image. Thus, Lie
subgroups are injectively immersed subgroups that are not necessarily closed.
Recall that a Lie group G acts properly on a manifold N if the map G × N → N × N sending
(g, x) to (x, g · x) is proper. The action is said to have constant orbit-type if all stabilisers are
conjugate. If a Lie group acts properly and with a constant orbit type, then the quotient is
a manifold and the quotient map is a fibre bundle.
Lemma 4.1 (quotient in stages). Let a Lie group G act on a manifold N . Let K be a Lie
subgroup of G that is normal in G. Also consider the induced action of G on the quotient N/K.
(i) There exists a unique map e : N/G → (N/K)/G such that the following diagram commutes:
N
πG
N/G
πK
e
/ N/K
πG/K
/ (N/K)/G
(4.1)
Basic Forms and Orbit Spaces: a Diffeological Approach
9
(ii) The map e is a diffeomorphism.
(iii) The pullback map
∗
πK
: Ω∗ (N/K) → Ω∗ (N )
∗
∗.
restricts to a bijection from Image πG/K
onto Image πG
(iv) Suppose that K acts on N properly and with a constant orbit-type, so that N/K is a mani∗ also restricts to a bijection
fold and N → N/K is a fibre bundle. Then the pullback map πK
∗
∗
∗
from Ωbasic (N/K) onto Ωbasic (N ). Consequently, if one of the inclusions Image πG/K
⊂
∗
∗
∗
Ωbasic (N/K) and Image πG ⊂ Ωbasic (N ) of Proposition 3.2 is an equality, then so is the
other.
Remark 4.2. The G-action on N/K factors through an action of G/K. The quotients (N/K)/G
and (N/K)/(G/K) coincide, as they are quotients of N/K by the same equivalence relation.
If K is closed in G (so that G/K is a Lie group) and N/K is a manifold, then G-basic forms
coincide with (G/K)-basic forms on N/K.
Proof of Lemma 4.1. Because πK is G-equivariant, such a map e exists. Because πG is onto,
such a map e is unique. Because the preimage under πK of a G-orbit in N/K is a single G-orbit
in N , the map e is one-to-one. Because the maps πG/K and πK are onto, the map e is onto.
Thus, the map e is a bijection. To show that it is a diffeomorphism, it remains to show,
for every parametrisation p : U → N/G, that p is a plot of N/G if and only if e ◦ p is a plot of
(N/K)/G.
Fix a parametrisation,
p : U → N/G.
Suppose that p is a plot of N/G. Let u ∈ U . Then there exist an open neighbourhood W of u
in U and a plot q : W → N such that p|W = πG ◦ q. The composition πK ◦ q is a plot of N/K,
and
πG/K ◦ πK ◦ q = e ◦ πG ◦ q = e ◦ p|W .
Because u ∈ U was arbitrary, this shows that e ◦ p is a plot of (N/K)/G.
Conversely, suppose that e ◦ p is a plot of (N/K)/G. Let u ∈ U . By applying the definition
of the quotient diffeology at πG/K and then at πK , we obtain an open neighbourhood W of u
in U and a plot r : W → N such that
e ◦ p|W = πG/K ◦ πK ◦ r.
By (4.1) and by the choice of r,
e ◦ πG ◦ r = πG/K ◦ πK ◦ r = e ◦ p|W .
Because e is one-to-one, this implies that πG ◦ r = p|W . Because u ∈ U was arbitrary, this
shows that p is a plot of N/G. This completes the proof that e is a diffeomorphism.
∗ takes Image π ∗
∗
Because the diagram (4.1) commutes, πK
G/K into Image πG . Because e is
a diffeomorphism, we can consider its inverse. From the commuting diagram
N
πG
N/G o
πK
e−1
/ N/K
πG/K
(N/K)/G
10
Y. Karshon and J. Watts
∗ takes Image π ∗
∗
∗
we see that πK
G/K onto Image πG . Because πK is one-to-one (by Lemma 3.1), we
∗
∗
∗
have a bijection πK : Image πG/K → Image πG .
Now suppose that K acts on N properly and with a constant orbit-type, so that N/K is
∗ is a bijection from the differential
a manifold and πK is a fibre bundle. Then we know that πK
forms on N/K to the K-basic differential forms on N .
∗ takes G-invariant forms on N/K to G-invariant forms on
Because πK is G-equivariant, πK
N and G-horizontal forms on N/K to G-horizontal forms on N . So we have an injection
∗ : Ω∗
∗
πK
basic (N/K) → Ωbasic (N ).
Let α be a G-basic form on N . In particular α is K-basic, so there exists a form β on N/K
∗ β. Because π ∗ is one-to-one and α is G-invariant, β is G-invariant. Because
such that α = πK
K
πK is G-equivariant and α is G-horizontal, β is G-horizontal. This completes the proof that the
∗ : Ω∗
∗
map πK
basic (N/K) → Ωbasic (N ) is a bijection.
5
The pullback surjects onto basics forms
The main result of this section is Proposition 5.10, in which we give conditions on an action
of a Lie group G on a manifold M under which every basic form α is the pullback of some
diffeological form on the quotient. By Proposition 2.20, we need to show that p∗1 α = p∗2 α for
every two plots p1 : U → M and p2 : U → M such that for each u ∈ U there is some g ∈ G
such that p2 (u) = g · p1 (u). If g can be chosen to be a smooth function of u, then it is easy to
conclude that p∗1 α = p∗2 α if α is basic:
Lemma 5.1. Let a Lie group G act on a manifold M . Let p1 , p2 : U → M be plots. Suppose
that p2 (u) = a(u) · p1 (u) for some smooth function a : U → G. Then for every α ∈ Ωkbasic (M )
we have p∗1 α = p∗2 α.
Proof . Pick a point u∈ U and a tangent vector v ∈ Tu U . Let ξ1 = (p1 )∗ v ∈ Tp1 (u) M and
ξ2 = (p2 )∗ v ∈ Tp2 (u) M . Let g = a(u). The directional derivative Dv a|u of a(·) in the direction
of v has the form η · g (∈ Tg G) for some Lie algebra element η (i.e., it is the right translation
of η by g). We then have that
ξ2 = g · ξ1 + ηM |p2 (u) ,
where g ·ξ1 is the image of ξ1 under the differential (push-forward) map g∗ : Tp1 (u) M → Tp2 (u) M ,
and where ηM is the vector field on M that corresponds to η; in particular ηM is everywhere
tangent to the G orbits.
Applying this to vectors v (1) , . . . , v (k) ∈ Tu U , we get that
(1)
(k) (p∗2 α)|u v (1) , . . . , v (k) = α|p2 (u) ξ2 , . . . , ξ2
= α|p2 (u)
(j)
where ξ2 := (p2 )∗ v (j)
(1)
(1)
(k)
(k) g · ξ1 + ηM , . . . , g · ξ1 + ηM
(j)
where ξ1 := (p1 )∗ v (j) and Dv(j) a|u = η (j) · g
(1)
(k) = α|p2 (u) g · ξ1 , . . . , g · ξ1
because α is horizontal
(1)
(k)
= α|p1 (u) ξ1 , . . . , ξ1
because α is invariant
= (p∗1 α)|u v (1) , . . . , v (k) .
Unfortunately, in applying Proposition 2.20, it might be impossible to choose g to be a smooth
function of u:
Basic Forms and Orbit Spaces: a Diffeological Approach
11
Example 5.2 (Z2 R). Let M = R, let G = {1, −1} with (±1)·x = ±x, and let π : M → M/G
be the quotient map. Consider the two plots p1 : R → M and p2 : R → M defined as follows:

−1/t2 if t < 0,

−e
p1 (t) := 0
if t = 0,

 −1/t2
e
if t > 0,
and
(
2
−e−1/t
p2 (t) :=
0
if t 6= 0,
if t = 0.
Then π ◦ p1 = π ◦ p2 . However, for t < 0 we have p1 (t) = 1 · p2 (t), whereas for t > 0 we have
p1 (t) = −1 · p2 (t), and so the two plots do not differ by a continuous function to G on any
neighbourhood of t = 0.
Our proofs use the following lemma.
Lemma 5.3. Let U ⊆ Rn be an open set. Let {Ci } be a (finite or) countable Scollection of
relatively closed subsets of U whose union is U . Then the union of the interiors, i int(Ci ), is
open and dense in U .
Proof . This is a consequence of the Baire category theorem.
We now prove Proposition 5.10 in the special case of a finite group action. In this case, basic
differential forms are simply invariant differential forms, as the tangent space to an orbit at any
point is trivial.
Proposition 5.4 (case of a finite group). Let G be a finite group, acting on a manifold M .
Then every basic form on M is the pullback of a (diffeological) differential form on M/G.
Proof . Fix a basic differential k-form α on M . By Proposition 2.20, it is enough to show the
following: if p1 : U → M and p2 : U → M are plots such that π ◦ p1 = π ◦ p2 , then p∗1 α = p∗2 α
on U . Fix two such plots p1 : U → M and p2 : U → M . For each g ∈ G let
Cg := {u ∈ U | g · p1 (u) = p2 (u)}.
By continuity, Cg is closed for each g. By our assumption on p1 and p2 ,
[
U=
Cg .
g∈G
S
By Lemma 5.3, the set g∈G int(Cg ) is open and dense in U . Thus, by continuity, it is enough
to show that p∗1 α = p∗2 α on int(Cg ) for each g ∈ G. This, in turn, follows from the facts that,
for each g ∈ G, we have g ◦ p1 = p2 on int(Cg ) and g ∗ α = α.
Our next result, contained in Lemma 5.6 and preceded by Lemma 5.5, is a generalisation of
the case of a finite group action: it shows that the property that interests us holds for a Lie
group action if it holds for the action of the identity component of that Lie group, assuming
that the action of the identity component is proper.
Lemma 5.5. Let G be a Lie group, and let G0 be its identity component. Assume that G acts
on a manifold M such that the restricted action of G0 on M is proper. Then, for any γ ∈ G/G0 ,
and for any two plots p1 : U → M and p2 : U → M , the set
Cγ := {u ∈ U | ∃ g ∈ γ such that g · p1 (u) = p2 (u)}
is (relatively) closed in U .
12
Y. Karshon and J. Watts
Proof . Because the G0 action on M is proper, the set
∆ := {(m, m0 ) ∈ M × M | ∃ g0 ∈ G0 such that g0 · m = m0 },
being the image of the proper map (g0 , m) 7→ (m, g0 · m), is closed in M × M .
Fix g 0 ∈ γ. Because G0 is normal in G, we can express γ as the left coset G0 g 0 , and we have
Cγ = {u ∈ U | ∃ g0 ∈ G0 such that g0 g 0 · p1 (u) = p2 (u)}.
We conclude by noting that Cγ is the preimage of the closed set ∆ under the continuous map
U → M × M given by u 7→ (g 0 · p1 (u), p2 (u)).
Lemma 5.6. Let G be a Lie group. Let G0 be the identity component of G. Fix an action
of G on a manifold M . Suppose that the restricted G0 -action is proper, and suppose that every
G0 -basic differential form on M is the pullback of a diffeological form on M/G0 . Then every
G-basic differential form on M is the pullback of a diffeological form on M/G.
Proof . Fix a G-basic form α on M . Let π : M → M/G be the quotient map, and let p1 : U → M
and p2 : U → M be plots such that π ◦ p1 = π ◦ p2 . Fix γ ∈ G/G0 and g 0 ∈ γ, and define Cγ as
in Lemma 5.5. Define p̃1 : U → M as the composition g 0 ◦ p1 . This is a plot of M , and for any
u ∈ int(Cγ ) we have p2 (u) = g0 · p̃1 (u) for some g0 ∈ G0 . Consider the restricted action of G0
on M . Let π0 : M → M/G0 be the corresponding quotient map. Then the restrictions p̃1 |int(Cγ )
and p2 |int(Cγ ) are plots of M , and they satisfy π0 ◦ p̃1 |int(Cγ ) = π0 ◦ p2 |int(Cγ ) . By hypothesis, and
because α is G0 -basic (as it is G-basic), α is a pullback of a diffeological form on M/G0 . By
Proposition 2.20, p̃∗1 α = p∗2 α on int(Cγ ).
But on int(Cγ ) we have p̃∗1 α = p∗1 g 0 ∗ α S
= p∗1 α (since α is G-invariant), and so p∗1 α = p∗2 α on
int(Cγ ). Since γ ∈ G/G0 is arbitrary, and γ∈G/G0 int(Cγ ) is open and dense in U by Lemmas 5.3
and 5.5, from continuity we have that p∗1 α = p∗2 α on all of U . Finally, by Proposition 2.20, α is
the pullback of a form on M/G.
We proceed with two technical lemmas that we will use to handle non-trivial compact connected stabilisers.
Lemma 5.7. Let G be a compact connected Lie group acting orthogonally on some Euclidean
space V = RN . Let g ∈ G and η ∈ g be such that exp(η) = g. Let v ∈ V . Then there exists
v 0 ∈ V such that |v 0 | ≤ |v| and g · v − v = η · v 0 .
Proof . Since V is a vector space, we identify tangent spaces at points of V with V itself. We
also identify elements of the group G and of the Lie algebra g with the matrices by which these
elements act on V = RN
1
g · v − v = exp(tη) · v 0
Z 1
d
=
exp(tη) · v dt
dt
0
Z 1
=
(η · exp(tη) · v) dt
0
Z 1
=η·
(exp(tη) · v) dt.
0
Basic Forms and Orbit Spaces: a Diffeological Approach
So define v 0 :=
R1
0
13
(exp(tη) · v) dt. Finally,
Z 1
|v | = (exp(tη) · v) dt
0
Z 1
|exp(tη) · v| dt
≤
0
Z 1
|v|dt because the action is orthogonal
=
0
0
= |v|.
This completes the proof.
Lemma 5.8. Let G be a compact connected Lie group acting orthogonally on some Euclidean
space V = RN . Let γ1 and γ2 be smooth curves from R into V such that γ1 (0) = γ2 (0) = 0 and
such that for every t ∈ R there exists gt ∈ G satisfying γ2 (t) = gt · γ1 (t). Let ξ1 = γ̇1 (0) and
ξ2 = γ̇2 (0). Then, for every horizontal form α on V , we have (ξ2 − ξ1 )y α|0 = 0.
Note that, in the assumptions of this lemma, t 7→ gt is not necessarily continuous.
Proof . We claim that there exists a sequence of vectors vn0 converging to 0 in V , and a sequence µn in g, such that
ξ2 − ξ1 = lim µn · vn0 .
n→∞
Indeed, choose any sequence of non-zero real numbers tn converging to 0. For each n choose
ηn ∈ g such that exp(ηn ) = gtn . Since we are working on a vector space, we can subtract the
curves and consider γ2 (t) − γ1 (t). We have
d (γ2 (t) − γ1 (t))
dt t=0
γ2 (t) − γ1 (t)
= lim
t→0
t
γ2 (tn ) − γ1 (tn )
= lim
n→∞
tn
gtn · γ1 (tn ) − γ1 (tn )
= lim
n→∞
tn
0
ηn · vn
= lim
n→∞
tn
ξ2 − ξ1 =
for some ηn ∈ g and for some vn0 ∈ V that satisfy |vn0 | ≤ |γ1 (tn )|; the last equality is a result of
Lemma 5.7. Because |vn0 | ≤ |γ1 (tn )| → |γ1 (0)| = 0, the claim holds with µn := ηn /tn .
n→∞
We now have
(ξ2 − ξ1 )y α|0 = lim (µn · vn0 )y (α|vn0 )
n→∞
= lim (µn )V y α|vn0 ,
n→∞
where (µn )V is the vector field on V induced by µn ∈ g. Because α is horizontal, the last term
above vanishes.
14
Y. Karshon and J. Watts
The final ingredient that we need for Proposition 5.10 is the following property of the model
that appears in the slice theorem (Theorem 2.25). Let G be a Lie group, H a closed subgroup,
and V a vector space with a linear H-action. Recall that the equivariant vector bundle G ×H V
over G/H is obtained as the quotient of G × V by the anti-diagonal H-action h · (g, v) =
(gh−1 , h · v) and that the G-action on G ×H V is g · [g 0 , v] = [gg 0 , v].
Lemma 5.9. Suppose that every H-basic form on V is the pullback of a diffeological form
on V /H. Then every G-basic form on G ×H V is the pullback of a diffeological form on
(G ×H V )/G.
Proof . Let G × H act on G × V where G acts by left multiplication on the first factor and
where H acts by the anti-diagonal action h : (g, v) →
7 (gh−1 , h · v). We have two maps:
G×V
pr2
V
πH
&
{
G ×H V
Here, the map πH : G × V → G ×H V is the quotient by the H-action, and the projection to the
second factor pr2 : G × V → V can be identified with the quotient by the G-action.
We apply quotient in stages (Lemma 4.1) in two ways: taking the quotient by G and then
by H, and taking the quotient by H and then by G. This gives the following commuting diagram:
G×V
pr2
V
πV
v
πH
V /H
)
πG×H
e
/ (G × V )/(G × H) o
e0
G ×H V
(5.1)
π
(G ×H V )/G
where e and e0 are diffeomorphisms, and πV , πG×H , and π are the quotient maps.
By hypothesis, the inclusion Image πV∗ ⊂ Ω∗basic (V ) is an equality. Applying Part (iv)
of Lemma 4.1 to the left hand side of the diagram (5.1), we conclude that the inclusion
∗
⊂ Ω∗basic (G × V ) is an equality. Applying Part (iv) of Lemma 4.1 to the right hand
Image πG×H
side of the diagram (5.1), we further conclude that the inclusion Image π ∗ ⊂ Ω∗basic (G ×H V ) is
an equality.
We are now ready to prove the main result of this section.
Proposition 5.10 (pullback surjects to basic forms). Let a Lie group G act on a manifold M .
Assume that the identity component of G acts properly. Let π : M → M/G be the quotient map.
Then every basic form on M is the pullback of a (diffeological) differential form on M/G via π ∗ .
Proof . By Lemma 5.6, to prove this result for an arbitrary Lie group, it is enough to prove it
for the action of the identity component of the group.
For every non-negative integer d, consider the following two statements.
A(d): For every connected Lie group K with dim K = d, and for every K-manifold N on which
the K-action is proper, every K-basic form on N is the pullback of a differential form
on N/K.
B(d): For every Lie group K with dim K = d, and for every K-manifold N on which the identity
component of K acts properly, every K-basic form on N is the pullback of a differential
form on N/K.
Basic Forms and Orbit Spaces: a Diffeological Approach
15
Since the result holds for the trivial group, Statement A(0) is true. By Lemma 5.6, it follows
that Statement B(0) is true. Proceeding by induction, we fix a positive integer d, we assume
that Statement B(d0 ) is true for all d0 < d, and we would like to prove that Statement B(d) is
true. By Lemma 5.6, Statement A(d) implies Statement B(d); thus, it is enough to prove that
Statement A(d) is true. That is, we may now restrict to the special case of the proposition in
which the group is connected and the action is proper, while assuming that the general case of
the proposition is true for all Lie groups of smaller dimension.
Now, let G be a connected Lie group, and let M be a G-manifold on which the G-action is
proper. Fix a G-basic form α on M . We would like to show that α is the pullback of a differential
form on M/G.
By Proposition 2.20 we need to show, for any two plots p1 : W → M and p2 : W → M for
which π ◦ p1 = π ◦ p2 , that p∗1 α = p∗2 α. Let p1 and p2 be two such plots. Fix u ∈ W . We would
like to show that p∗1 α|u = p∗2 α|u .
Let x = p2 (u). Let H be the stabiliser of x. By Theorem 2.25 there exists a G-invariant
open neighbourhood U of x and an equivariant diffeomorphism F : U → G ×H V where V =
Tx M/Tx (G·x). Because F is an equivariant diffeomorphism and α is G-basic, (F −1 )∗ α is G-basic
on G ×H V .
Either x is a fixed point, or x is not a fixed point.
Suppose that x is a fixed point. Then H = G. So p1 (u) = p2 (u) = x, and F identifies U
with V = Tx M , sending x to 0 ∈ V . Fixing a G-invariant Riemannian metric, we have that G
acts linearly and orthogonally on V . Let v ∈ Tu W . Applying Lemma 5.8 to the curves γ1 (t) :=
F (p1 (u+tv)) and γ2 (t) := F (p2 (u+tv)) in V and to the basic form (F −1 )∗ α on V , we obtain that
γ̇1 (0)y (F −1 )∗ α|0 = γ̇2 (0)y (F −1 )∗ α|0 . This, in turn, implies that vy p∗1 α|u = vy p∗2 α|u . Because
v ∈ Tu W is arbitrary, we conclude that p∗1 α|u = p∗2 α|u , as required.
Suppose that x is not a fixed point. Then the stabiliser H of x is a proper subgroup of G.
Since G is connected, dim H < dim G. By the induction hypothesis, every H-basic form on V
is the pullback of a differential form on V /H. By Lemma 5.9, every G-basic form on G ×H V is
the pullback of a differential form on (G ×H V )/G. Because F is an equivariant diffeomorphism,
every G-basic form on U is the pullback of a differential form on U/G. So α|U is the pullback
of a differential form on U/G. This implies that p∗1 α|u = p∗2 α|u , as required.
Proof of Theorem 1.1. By Lemma 3.1 and Proposition 3.2, the pullback is an injection into
the space of basic forms. By Remark 2.17 and Proposition 3.4, as a map to its image, the
pullback is an isomorphism of differential graded algebras and a diffeological diffeomorphism.
By Proposition 5.10, if the identity component of G acts properly, the image is the space of
basic forms.
Example 5.11 (irrational torus, first construction). Fix an irrational number α ∈ RrQ. The
corresponding irrational torus is
Tα := R/(Z + αZ).
It is obtained as the quotient of R by the Z2 -action (m, n) · x = x + m + nα; note that it is
not Hausdorff. The basic differential forms on R with respect to this action are the constant
functions and the constant coefficient one-forms cdx. By Proposition 5.10, each of these is the
pullback of a differential form on Tα .
Remark 5.12. In Example 5.11, although the topology of Tα is trivial, its de Rham cohomology
is isomorphic to that of a circle. We note, though, that differential forms still do not capture
the richness of the diffeology on Tα : by Donato and Iglesias [8], Tα and Tβ are diffeomorphic if
and only if there exist integers a, b, c, d such that ad − bc = ±1 and α = a+βb
c+βd . See Exercise 4
and Exercise 105 of [14] with solutions at the end of the book.
16
Y. Karshon and J. Watts
Example 5.13 (irrational torus, second construction). Fix an irrational number α ∈ RrQ.
Consider the quotient
T2 /Sα
of T2 := R2 /Z2 by the irrational solenoid Sα := {[t, αt] | t ∈ R} ⊂ T2 . It is obtained as the
quotient of T2 by the R-action t · [x, y] = [x + t, y + αt]. The basic forms on T2 with respect
to this action are the constant functions and the constant multiples of the one-form αdx − dy.
The quotient T2 /Sα is diffeomorphic to the irrational torus Tα of Example 5.11; see Exercise 31
of [14] (with solution at the end of the book).
In fact, consider the action of R×Z2 on R2 that is given by (t, m, n)·(x, y) = (x+m+t, y +n+
αt). Taking the quotient first by R and then by Z2 (and identifying the first of these quotients
with R through the map (x, y) 7→ y − αx) yields Tα . Taking the quotient first by Z2 and then
by R yields T2 /Sα . Applying Lemma 4.1 twice, we get the following commuting diagram:
R2
R
Tα
'
x
e
/ R2 /(R × Z2 ) o
e0
T2
T2 /Sα
where e and e0 are diffeomorphisms. As noted in Example 5.11, every basic form on R is
the pullback of a diffeological differential form on Tα . By Lemma 4.1, this implies that every
basic form on R2 is the pullback of a diffeological differential form on R2 /(R × Z2 ). Again by
Lemma 4.1, we conclude that every basic form on T2 is the pullback of a diffeological form
on T2 /Sα . Thus, the R-action on T2 through Sα satisfies the conclusion of Proposition 5.10,
although it does not satisfy the assumption of Proposition 5.10: this R-action is not proper.
A
Orbifolds
Let X be a Hausdorff, second countable topological space. Fix a positive integer n.
The following definition is based on Haefliger, [11, Section 4].
1. An n dimensional orbifold chart on X is a triple (Ũ , Γ, φ) where Ũ ⊆ Rn is an open ball,
Γ is a finite group of diffeomorphisms of Ũ , and φ : Ũ → X is a Γ-invariant map onto an
open subset U of X that induces a homeomorphism Ũ /Γ → U .
2. Two orbifold charts on X, (Ũ , Γ, φ) and (Ṽ , Γ0 , ψ), are compatible if for every two points
u ∈ Ũ and v ∈ Ṽ such that φ(u) = ψ(v) there exist a neighbourhood Ou of u in Ũ and
a neighbourhood Ov of v in Ṽ and a diffeomorphism g : Ou → Ov that takes u to v and
such that ψ ◦ g = φ.
3. An orbifold atlas on X is a set of orbifold charts on X that are pairwise compatible and
whose images cover X. Two orbifold atlases are equivalent if their union is an orbifold atlas.
The following definition was introduced in [13]: A diffeological orbifold is a diffeological space
that is locally diffeomorphic to finite linear quotients of Rn .
These two definitions are equivalent in the following sense. Given an orbifold atlas on X, there
exists a unique diffeology on X such that all the homeomorphisms Ũ /Γ → U are diffeomorphisms. With this diffeology, X becomes a diffeological orbifold. Two orbifold atlases are
Basic Forms and Orbit Spaces: a Diffeological Approach
17
equivalent if and only if the corresponding diffeologies are the same. Finally, every diffeological
orbifold structure on X can be obtained in this way. For details, see [13, Section 8].
Orbifolds were initially introduced by Ichiro Satake [20, 21] under the name “V-manifolds”.
Satake’s approach is equivalent to Haefliger’s; see [13]. Satake [21] also introduced tensors,
and in particular differential forms, on V-manifolds. Haefliger’s approach yields the following
definition.
e , Γ, ψ)} be an orbifold atlas on X. An orbifold differential form on X is given by, for
Let {(U
e , Γ, ψ) in the atlas, a Γ-invariant differential form α e on the domain U
e of the chart.
each chart (U
U
e , Γ, φ), (Ṽ , Γ0 , ψ), and
We require the following compatibility condition. For every two charts (U
e and v ∈ Ṽ with φ(u) = ψ(v), there exist a diffeomorphism g : Ou → Ov
every two points u ∈ U
from a neighbourhood of u to a neighbourhood of v that takes u to v, such that ψ ◦ g = φ, and
such that g ∗ (αṼ |Ov ) = αUe |Ou . Two such collections {αUe } of differential forms, defined on the
domains of the charts in two equivalent orbifold atlases, represent the same orbifold differential
form if their union still satisfies the compatibility condition.
Every diffeological differential form α on X determines an orbifold differential form by ase , Γ, ψ) the pullback ψ ∗ α. Proposition 5.4 implies that this gives
sociating to every chart (U
a bijection between diffeological differential forms and orbifold differential forms.
B
Sjamaar dif ferential forms;
case of regular symplectic quotients
Let a Lie group G act properly on a symplectic manifold (M, ω) with an (equivariant) momentum
map Φ : M → g∗ . Let Z = Φ−1 (0) be the zero level set and i : Z → M its inclusion map. Let
Zreg = {z ∈ Z | ∃ neighbourhood U of z in Z such that, for all z 0 ∈ U ,
the stabilisers of z 0 and of z are conjugate in G}.
The set Zreg , (with the subset diffeology induced from M or, equivalently, from Z) is a manifold,
and it is open and dense in Z (see [24]). The quotient Zreg /G, (with the quotient diffeology
induced from Zreg , or, equivalently, the subset diffeology induced from M/G,) is also a manifold.
Above, the connected components of Zreg and Zreg /G may have different dimensions. If M is
connected and Φ is proper, then Zreg and Zreg /G are connected. See, for example, [18] and [24].
Denote by ireg : Zreg → M the inclusion map and by πreg : Zreg → Zreg /G the quotient map.
The following definition was introduced (but not yet named) by Reyer Sjamaar in [23]:
Definition B.1. A Sjamaar differential l-form σ on Z/G is a differential l-form on Zreg /G (in
∗ σ.
the ordinary sense) such that there exists σ̃ ∈ Ωl (M ) satisfying i∗reg σ̃ = πreg
∗ ω
A special case of a Sjamaar form is the reduced symplectic form, ωred , which satisfies πreg
red =
The orbit type stratification on M induces a stratification of the reduced space Z/G, and
the Sjamaar differential forms naturally extend to the strata of Z/G. The extensions of ωred to
these strata exhibit Z/G as a stratified symplectic space in the sense of Sjamaar and Lerman [24].
The space of Sjamaar forms is closed under wedge products and forms a subcomplex of the
de Rham complex (Ω∗ (Zreg /G), d). Sjamaar forms satisfy a Poincaré lemma, Stokes’ theorem,
and a de Rham theorem.
For details, see Sjamaar’s paper [23].
The reduced space Z/G comes equipped with the quotient diffeology inherited from Z, which
equals the subset diffeology inherited from M/G. We call this the subquotient diffeology.
It is now natural to ask how Sjamaar forms on a symplectic quotient Z/G, which a-priori
depend on the ambient symplectic manifold M , relate to the diffeological forms on Z/G, whose
i∗reg ω.
18
Y. Karshon and J. Watts
definition is intrinsic. More precisely, consider the inclusion map J : Zreg /G → Z/G. Then we
have the pullback map on diffeological forms
J ∗ : Ωl (Z/G) → Ωl (Zreg /G),
and we identify the target space with the ordinary differential forms on Zreg /G. We ask:
• Is the space of Sjamaar forms contained in the image of J ∗ ?
• Is the image of J ∗ contained in the space of Sjamaar forms?
• Is J ∗ one-to-one?
If 0 is a regular value of the momentum map Φ, then it follows from Proposition 5.4 that
the answers to each of these questions is “yes”. If 0 is a critical value, then the answer to the
first question is “yes”, and we do not know the answers to the other two questions. We refer
the reader to Section 3.4 of the second author’s thesis [28] for details.
Acknowledgements
This work is partially supported by the Natural Sciences and Engineering Council of Canada. We
are grateful to Patrick Iglesias-Zemmour for instructing us on diffeology and to Reyer Sjamaar
for his inspiration, as well as to the anonymous referees for excellent suggestions that lead to
a better organisation of the paper.
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